Let us take the three vertices as $(1,2,10)$.
Now the simplest case is when only $3$ vertices are involved , i.e. $(1,2,10)=1$ way
Now we make room for one more vertex $(1, \_ ,2,10)$ or $(1,2,\_,10)$ which can be filled in $7$ ways giving a total of $7*2$ ways
For one more vertex we have $7*6*3$ ways and so on till we have accompanied all the $7$ vertices and along with $2$ a total of $8$ vertices are there with total ordering of $7!*8$
So, to sum it : $1+7*2+7*6*3+7*6*5*4+\ldots+7!*8 $
$\quad \quad = 7P0*1+7P1*2+7P2*3+\ldots+7P7*8$